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In this paper, we prove the stability results for measure differential equations, considering more general conditions under the Lyapunov functionals and concerning the functions f and g. Moreover, we prove these stability results for the dynamic equations on time scales, using the correspondence between the solutions of these last equations and the solutions of the measure differential equations. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the Cauchy problem of the 2D Zakharov-Kuznetsov-Burgers (ZKB) equation, which has the dissipative term − ∂ x 2 u. This is known that the 2D Zakharov-Kuznetsov equation is well-posed in H s (R 2) for s > 1 / 2 , and the 2D nonlinear parabolic equation with quadratic derivative nonlinearity is well-posed in H s (R 2) for s ≥ 0. By using the Fourier restriction norm with dissipative effect, we prove the well-posedness for ZKB equation in H s (R 2) for s > − 1 / 2. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stability of Boltzmann equation with large external potential in T 3. For a class of initial data with large oscillations in L x , v ∞ around the local Maxwellian, we prove the existence of a global solution to the Boltzmann equation provided the initial perturbation is suitably small in L 2 -norm. The large time behavior of the Boltzmann solution with exponential decay rate is also obtained. This seems to be the first result on the perturbation theory of large-amplitude non-constant equilibriums for large-amplitude initial data. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the Cauchy problem of the multi-dimensional inhomogeneous incompressible magnetohydrodynamic equations with fractional dissipation. We prove the unique global strong solution with vacuum to the magnetohydrodynamic equations over the whole space. In addition, the large time decay rates of the corresponding strong solution are also obtained. Note that our result is proved without any smallness on the initial data. Moreover, the initial density is allowed to be nonnegative, and in particular, the initial vacuum is allowed. To our best knowledge, this is the first result on the inhomogeneous fractional magnetohydrodynamic equations. [ABSTRACT FROM AUTHOR]

In the paper we consider the solution of an advection equation with rapidly changing coefficients ∂ t u ε + (1 / ε) V (t / ε 2 , x / ε) ⋅ ∇ x u ε = 0 for t < T and u ε (T , x) = u 0 (x) , x ∈ R d. Here ε > 0 is some small parameter and the drift term (V (t , x)) (t , x) ∈ R 1 + d is assumed to be a d -dimensional, vector valued random field with incompressible spatial realizations. We prove that when the field is Gaussian, locally stationary, quasi-periodic in the x variable and strongly mixing in time the solutions u ε (t , x) converge in law, as ε → 0 , to u 0 (x (T ; t , x)) , where (x (s ; t , x)) s ≥ t is a diffusion satisfying x (t ; t , x) = x. The averages of u ε (T , x) converge then to the solution of the corresponding Kolmogorov backward equation. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study a class of semilinear ill-posed equations with sectorially dichotomous operator S on Banach space Z. Firstly we give a direct sum decomposition of Z , Z + ⊕ Z − = Z , corresponding to spectrum of S such that hyperbolic bisectorial operator S can be split into two sectorial operators S | Z + and − S | Z − on Z + and Z − , respectively. Then we construct the intermediate spaces between whole space Z and domain D (S) of sectorially dichotomous operator S. Following ElBialy's works, we propose the dichotomous initial condition for this semilinear ill-posed equation, and obtain the existence, uniqueness, continuous dependence on the dichotomous initial value, regularity and Z α -estimate of dichotomous solutions. As applications of the results, we give the existence and uniqueness of local solutions for an elliptic PDE in infinite cylindrical domain and an abstract semilinear ill-posed equation with non-dense domain. [ABSTRACT FROM AUTHOR]

Abstract In this paper we use variational techniques to give existence results for the problem { S k [ u ] = f (x , − u) in Ω u < 0 in Ω u = 0 on ∂ Ω where S k [ u ] is the k -Hessian operator and f (x , u) is a supercritical nonlinearity in the sense introduced by [K. Tso, Ann. Inst. Henri Poincaré (1990)]. Using some ideas from a celebrated article by Brezis and Nirenberg we show existence of a positive solution considering supercritical nonlinearities, which is surprising given the validity of the Pohozaev identity. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the following singular Monge–Ampère equation (0.1) { det ⁡ D 2 u = 1 (H u) n + k + 2 u ⁎ k in Ω ⊂ ⊂ R n , u = 0 , on ∂ Ω where k ≥ 0 , H < 0 are constants and u ⁎ = x ⋅ ∇ u (x) − u (x) is the Legendre transformation of u. Equation (0.1) is related to proper affine hyperspheres. We will show the existence of solutions of (0.1) u ∈ C ∞ (Ω) ∩ C (Ω ¯) via regularization method. Using the technique in [10,12] , we also obtain the optimal graph regularity of the solution of (0.1). [ABSTRACT FROM AUTHOR]

Abstract The paper concerns existence and uniqueness of solutions to a nonlinear parabolic equation with merely integrable data on a Lipschitz bounded domain in R N. Our focal point is to involve the leading, nonlinear part of the operator whose growth is described by anisotropic N -function M inhomogeneous in the space and the time variables. The main goals are proven in absence of Lavrentiev's phenomenon, to ensure which we impose a certain type of balance of interplay between the behavior of M for large | ξ | and small changes of time and space variables. Its instances are log-Hölder continuity of variable exponent (inhomogeneous in time and space) or optimal closeness condition for powers in double phase spaces (possibly changing in time). New delicate approximation-in-time result is proven and applied in the construction of renormalized solutions. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the global stability of homogeneous equilibria in Keller–Segel–Navier–Stokes equations in scaling-invariant spaces. We prove that for any given 0 < M < 1 + μ 1 with μ 1 being the first eigenvalue of Neumann Laplacian, the initial–boundary value problem of the Keller–Segel–Navier–Stokes system has a unique globally bounded classical solution provided that the initial datum is chosen sufficiently close to (M , M , 0) in the norm of L d / 2 (Ω) × W ˙ 1 , d (Ω) × L d (Ω) and satisfies a natural average mass condition. Our proof is based on the perturbation theory of semigroups and certain delicate exponential decay estimates for the linearized semigroup. Our result suggests a new observation that nontrivial classical solution for Keller–Segel–Navier–Stokes equation can be obtained globally starting from suitable initial data with arbitrarily large total mass provided that volume of the bounded domain is large, correspondingly. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study orbital stability of peakons for the generalized modified Camassa–Holm (gmCH) equation, which is a natural higher-order generalization of the modified Camassa–Holm (mCH) equation, and admits Hamiltonian form and single peakons. We first show that the single peakon is the usual weak solution of the PDEs. Some sign invariant properties and conserved densities are presented. Next, by constructing the corresponding auxiliary function h (t , x) and establishing a delicate polynomial inequality relating to the two conserved densities with the maximal value of approximate solutions, the orbital stability of single peakon of the gmCH equation is verified. We introduce a new approach to prove the key inequality, which is different from that used for the mCH equation. This extends the result on the stability of peakons for the mCH equation (Qu et al. 2013) [36] successfully to the higher-order case, and is helpful to understand how higher-order nonlinearities affect the dispersion dynamics. [ABSTRACT FROM AUTHOR]

Abstract We develop a new and general method to prove the existence of the random attractor (strong attractor) for the primitive equations (PEs) of large-scale ocean and atmosphere dynamics under non-periodic boundary conditions and driven by infinite-dimensional additive fractional Wiener processes. In contrast to our new method, the common method, compact Sobolev embedding theorem, is to obtain the time-uniform a priori estimates in some Sobolev space whose regularity is higher than the solution space. But this method can not be applied to the 3D stochastic PEs with the non-periodic boundary conditions. Therefore, the existence of universal attractor (weak attractor) was established in previous works (see [15,16]). The main idea of our method is that we first derive that P -almost surely the solution operator of stochastic PEs is compact. Then we construct a compact absorbing set by virtue of the compact property of the solution operator and the existence of a absorbing set. We should point out that our method has some advantages over the common method of using compact Sobolev embedding theorem, i.e., using our method we only need to obtain time-uniform a priori estimates in the solution space to prove the existence of random attractor for the corresponding stochastic system, while the common method need to establish time-uniform a priori estimates in a more regular functional space than the solution space. Take the stochastic PEs for example, as the unique strong solution to the stochastic PEs belongs to C ([ 0 , T ] ; (H 1 (℧)) 3) , in view of our method, we only need to obtain the time-uniform a priori estimates in the solution space (H 1 (℧)) 3 to prove the existence of random attractor for this stochastic system, while the common method need to establish time-uniform a priori estimates for the solution in the functional space (H 2 (℧)) 3. However, time-uniform a priori estimates in (H 2 (℧)) 3 for the solution to stochastic PEs are too difficult to be established. The present work provides a general way for proving the existence of random attractor for common classes of dissipative stochastic partial differential equations driven by Wiener noises, fractional noises and even jump noises. In a forth coming paper, using this new method we [46] prove the existence of random attractor for the stochastic nematic liquid crystals equations. This is the first result about the long-time behavior of stochastic nematic liquid crystals equations. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the existence and concentration behavior of minimizers for i V (c) = inf u ∈ S c ⁡ I V (u) , here S c = { u ∈ H 1 (R N) | ∫ R N V (x) | u | 2 < + ∞ , | u | 2 = c > 0 } and I V (u) = 1 2 ∫ R N (a | ∇ u | 2 + V (x) | u | 2) + b 4 (∫ R N | ∇ u | 2) 2 − 1 p ∫ R N | u | p , where N = 1 , 2 , 3 and a , b > 0 are constants. By the Gagliardo–Nirenberg inequality, we get the sharp existence of global constraint minimizers of i V (c) for 2 < p < 2 ⁎ when V (x) ≥ 0 , V (x) ∈ L ∞ l o c (R N) and lim | x | → + ∞ ⁡ V (x) = + ∞. For the case p ∈ (2 , 2 N + 8 N) \ { 4 } , we prove that the global constraint minimizers u c of i V (c) behave like u c (x) ≈ c | Q p | 2 ( m c c) N 2 Q p ( m c c x − z c) , for some z c ∈ R N when c is large, where Q p is, up to translations, the unique positive solution of − N (p − 2) 4 Δ Q p + 2 N − p (N − 2) 4 Q p = | Q p | p − 2 Q p in R N and m c = ( a 2 D 1 2 − 4 b D 2 i 0 (c) + a D 1 2 b D 2 ) 1 2 , D 1 = N p − 2 N − 4 2 N (p − 2) and D 2 = 2 N + 8 − N p 4 N (p − 2). [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study a weakly dissipative Dullin–Gottwald–Holm equation from the viewpoint of Lie symmetry analysis. We first perform symmetry analysis and the nonlinear self-adjointness of this equation. Due to a mixed derivatives term in the equation, we need to rewrite the corresponding form Lagrangian in symmetric form to construct conservation laws. From the viewpoint, we present a general procedure of how these conserved quantities come about. Based on these conserved quantities, blow-up analysis and global existence of strong solutions are presented. Finally, we show that this equation admits a weak peakon-type solution. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the fractional Hardy–Hénon equations with an isolated singularity. If the isolated singularity is located at the origin, we give a classification of solutions to this equation. If the isolated singularity is located at infinity, in the case of exterior domains, we provide decay estimates of solutions and their gradients at infinity. Our results are an extension of the classical work by Caffarelli, Gidas et al. [ABSTRACT FROM AUTHOR]

In this paper, the wave breaking of the Fornberg–Whitham equation is studied. We first investigate the local well-posedness and show a blow-up scenario by local-in-time a priori estimates. Next, we present a sufficient condition on the initial data to ensure wave-breaking. Moreover, the estimates of life span and blow-up rate are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we find some new patterns regarding the periodic solvability of the Brillouin electron beam focusing equation x ¨ + β ( 1 + cos ⁡ ( t ) ) x = 1 x . In particular, we prove that there exists β ⁎ ≈ 0.248 for which a 2 π -periodic solution exists for every β ∈ ( 0 , β ⁎ ] , and the bifurcation diagram with respect to β displays a fold for β = β ⁎ . This result significantly contributes to the discussion about the well-known conjecture asserting that the Brillouin equation admits a periodic solution for every β ∈ ( 0 , 1 / 4 ) , leading to doubt about its truthfulness. For the first time, moreover, we prove multiplicity of periodic solutions for a range of values of β near β ⁎ . The technique used relies on rigorous computation and can be extended to some generalizations of the Brillouin equation, with right-hand side equal to 1 / x p ; we briefly discuss the cases p = 2 and p = 3 . [ABSTRACT FROM AUTHOR]

In this paper, we discuss several results regarding existence, non-existence and asymptotic properties of solutions to u ⁗ + q u ″ + f ( u ) = 0 , under various hypotheses on the parameter q and on the potential F ( t ) = ∫ 0 t f ( s ) d s , generally assumed to be bounded from below. We prove a non-existence result in the case q ≤ 0 and an existence result of periodic solution for: 1) almost every suitably small (depending on F ), positive values of q ; 2) all suitably large (depending on F ) values of q . Finally, we describe some conditions on F which ensure that some (or all) solutions u q to the equation satisfy ‖ u q ‖ ∞ → 0 , as q ↓ 0 . [ABSTRACT FROM AUTHOR]

In this paper we study the following Cauchy problem: u t = u x x + ( u n ) x , ( x , t ) ∈ R × ( 0 , ∞ ) , u ( x , 0 ) = 0 , x ≠ 0 , where parameter n ≥ 0 . Its nonnegative solution is called singular solution when u ( x , t ) satisfies the equation in the sense of distribution, initial conditions in the classical sense and also u ( x , t ) exhibits a singularity at the origin ( 0 , 0 ) . As we know, the singular solution is called source-type solution if the initial is M δ ( x ) , where δ ( x ) is Dirac measure and constant M > 0 . The solution is called very singular solution if it possesses more singularity than that of source-type solution at the origin. Here we focus on what happens in the interactive effect between the diffusion and convection in a whole physical process. We find critical values n 2 < n 1 < n 0 such that there exists unique source-type solution in the exponent range of 0 < n < n 0 , while there exists no nonnegative singular solution if n ≥ n 0 . Only in the case of n 2 < n < n 1 there exists a very singular solution, but in the case of n ≥ n 1 or n ≤ n 2 there is no solution that exhibits more singular than source-type solution at the origin. Furthermore we describe the short time asymptotic behavior of the singular solutions when such Cauchy problem is solvability for source-type solution or very singular solution. [ABSTRACT FROM AUTHOR]

In this paper, we study the blow-up mechanism and persistence property of solutions to the modified b-family of equations. The dynamics of the blow-up quantity along the characteristics is established by the Riccati-type differential inequality with various parameters. The key feature of the method is to refine the analysis on the growth rate of the relative ratio between solution and its gradient by performing a vertical shift. Furthermore, the persistence results for the solution are established in weighted spaces. [ABSTRACT FROM AUTHOR]

This paper deals with the limiting behavior of stochastic reaction–diffusion equations driven by multiplicative noise and deterministic non-autonomous terms defined on thin domains. We first prove the existence, uniqueness and periodicity of pullback tempered random attractors for the equations in an ( n + 1 ) -dimensional narrow domain, and then establish the upper semicontinuity of these attractors when a family of ( n + 1 ) -dimensional thin domains collapses onto an n -dimensional domain. [ABSTRACT FROM AUTHOR]

In this paper, we consider the periodic discrete nonlinear equation { L u n − ω u n = ± g n ( u n ) , n ∈ Z , lim | n | → ∞ ⁡ u n = 0 , where L is a Jacobi operator, and the nonlinearities g n ( s ) are asymptotically linear as | s | → ∞ . In the two different cases ( ω is a spectral endpoint of L , or it belongs to a finite spectral gap of L ), we obtain the existence of nontrivial solitons of this equation by using variational methods. In particular, a necessary and sufficient condition is obtained for the existence of gap solitons of the nonlinear equation. Here, solitons appear when we look for standing waves of some discrete nonlinear Schrödinger equations. [ABSTRACT FROM AUTHOR]

Abstract: The aims of this paper are to discuss existence and uniqueness of local solutions for a class of non-Newtonian fluids with singularity and vacuum in one-dimensional bounded intervals. There are two important points in this paper, one is that we allow the initial vacuum; another one is that the viscosity term of momentum equation is with singularity and fully nonlinearity. [Copyright &y& Elsevier]

Abstract: This paper is addressed to some questions concerning the exponential stability and its robustness measure for linear time-varying differential-algebraic systems of index 1. First, the Bohl exponent theory that is well known for ordinary differential equations is extended to differential-algebraic equations. Then, it is investigated that how the Bohl exponent and the stability radii with respect to dynamic perturbations for a differential-algebraic system depend on the system data. The paper can be considered as a continued and complementary part to a recent paper on stability radii for time-varying differential-algebraic equations [N.H. Du, V.H. Linh, Stability radii for linear time-varying differential-algebraic equations with respect to dynamic perturbations, J. Differential Equations 230 (2006) 579–599]. [Copyright &y& Elsevier]

Abstract: In a previous paper we introduced various definitions of stability and instability for non-autonomous differential equations, and applied these to investigate the bifurcations in some simple models. In this paper we present a more systematic theory of local bifurcations in scalar non-autonomous equations. [Copyright &y& Elsevier]

In this paper we extend previous results on the regularity of solutions of integro-differential parabolic equations. The kernels are non-necessarily symmetric which could be interpreted as a non-local drift with the same order as the diffusion. We provide a growth lemma and a Harnack inequality which can be used to prove higher regularity estimates. [ABSTRACT FROM AUTHOR]

The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger–Wang, Chau–Weinkove and the author solved this global problem in W 4 , p under some restrictions on the sign or integrability of the affine mean curvature. We remove these restrictions in this paper and obtain W 4 , p solution to the second boundary value problem when the affine mean curvature belongs to L p with p greater than the dimension. Our self-contained analysis also covers the case of Abreu's equation. [ABSTRACT FROM AUTHOR]

This paper studies traveling fronts to cooperation–diffusion systems in R N for N ≥ 3 . We consider ( N − 2 ) -dimensional smooth surfaces as boundaries of strictly convex compact sets in R N − 1 , and define an equivalence relation between them. We prove that there exists a traveling front associated with a given surface and show its stability. The associated traveling fronts coincide up to phase transition if and only if the given surfaces satisfy the equivalence relation. [ABSTRACT FROM AUTHOR]

Abstract: The paper deals with Monge–Kantorovich equation (MK for short) in an open bounded domain Ω with Dirichlet boundary condition. We study existence and uniqueness of a solution to the associated evolution problem (EMK for short) and we prove the convergence to a solution of MK, when time goes to ∞. A solution is a couple , where u is the potential and Φ is the transportation flux. We study the problem for a given Radon measure source term and we show how to use the numerical method of Dumont and Igbida (2009) [22] to provide numerical approximation of the solution of MK. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the chaotic behavior of ordinary differential equations with a homoclinic orbit to a saddle fixed point under an unbounded random forcing driven by a Brownian motion. We prove that, for almost all sample paths of the Brownian motion in the classical Wiener space, the forced equation admits a topological horseshoe of infinitely many branches. This result is then applied to the randomly forced Duffing equation and the pendulum equation. [Copyright &y& Elsevier]

Abstract: The main purpose of this paper is to justify the Stokes–Blasius law of boundary-layer thickness for the 2-D Boussinesq equations with vanishing diffusivity limit in the half plane, i.e., we shall prove that the boundary-layer thickness is of the value with any for small diffusivity coefficient . Moreover, the convergence rates of the vanishing diffusivity limit are also obtained. [Copyright &y& Elsevier]

Abstract: In this paper, we consider global subsonic compressible flows through an infinitely long axisymmetric nozzle. The flow is governed by the steady Euler equations and has boundary conditions on the nozzle walls. Existence and uniqueness of global subsonic solution are established for an infinitely long axisymmetric nozzle, when the variation of Bernoulli''s function in the upstream is sufficiently small and the mass flux of the incoming flow is less than some critical value. The results give a strictly mathematical proof to the assertion in Bers (1958) : there exists a critical value of the incoming mass flux such that a global subsonic flow exists uniquely in a nozzle, provided that the incoming mass flux is less than the critical value. The existence of subsonic flow is obtained by the precisely a priori estimates for the elliptic equation of two variables. With the assumptions on the nozzle in the far fields, the asymptotic behavior can be derived by a blow-up argument for the infinitely long nozzle. Finally, we obtain the uniqueness of uniformly subsonic flow by energy estimate and derive the existence of the critical value of incoming mass flux. [Copyright &y& Elsevier]

Abstract: In this paper, we use Borel''s procedure to construct Gevrey approximate solutions of an initial value problem for involutive systems of Gevrey complex vector fields. As an application, we describe the Gevrey wave-front set of the boundary values of approximate solutions in wedges of Gevrey involutive structures . We prove that the Gevrey wave-front set of the boundary value is contained in the polar of a certain cone contained in where X is a maximally real edge of . We also prove a partial converse. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with the existence and asymptotic behavior of standing wave solutions of nonlinear Schrödinger equations with electromagnetic fields , , with E being a critical frequency in the sense that . We show that if the zero set of has several isolated connected components such that the interior of is not empty and is smooth, then for large there exists, for any non-empty subset , a standing wave solution which is trapped in a neighborhood of . [Copyright &y& Elsevier]

Abstract: In this paper, we prove that the Cauchy problem for the nonlinear pseudo-parabolic equation admits a unique global generalized solution in , a unique global classical solution and asymptotic behavior of the solution. We also prove that the Cauchy problem for the equation has a unique global generalized solution in , a unique global classical solution and asymptotic behavior of the solution. [Copyright &y& Elsevier]

Abstract: In this paper, we study the Cauchy problem for the generalized finite-depth-fluid equation , where , k is an integer that is larger than 4. We obtain that it is globally wellposed if and the initial data in are sufficiently small, where and . Moreover, we show that its solution will converge to those of the generalized BO and KdV equations as the depth parameter and , respectively. [Copyright &y& Elsevier]

Abstract: In this paper we study the smoothness properties of solutions to the KP-I equation. We show that the equation''s dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data ϕ possesses certain regularity and sufficient decay as , then the solution will be smoother than ϕ for where T is the existence time of the solution. [Copyright &y& Elsevier]

Abstract: In this paper, we study the “triply” degenerate problem: on , on Ω and “ on some part of the boundary ,” in the case of continuous nonhomogeneous and nonstationary boundary data a. The functions are assumed to be continuous, locally Lipschitz, nondecreasing and to verify the normalization condition and the range condition . Using monotonicity and penalization methods, we prove existence of a weak renormalized entropy solution in the spirit of [K. Ammar, J. Carrillo, P. Wittbold, Scalar conservation laws with general boundary condition and continuous flux function, J. Differential Equations 228 (2006) 111–139]. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to study the reflection–transmission of diffractive geometrical optic rays described by semi-linear symmetric hyperbolic systems such as the Maxwell–Lorentz equations with the anharmonic model of polarization. The framework is that of P. Donnat''s thesis [P. Donnat, Quelques contributions mathématiques en optique non linéaire, chapters 1 and 2, thèse, 1996] and V. Lescarret [V. Lescarret, Wave transmission in dispersive media, M3AS 17 (4) (2007) 485–535]: we consider an infinite WKB expansion of the wave over long times/distances and because of the boundary, we decompose each profile into a hyperbolic (purely oscillating) part and elliptic (evanescent) part as in M. William [M. William, Boundary layers and glancing blow-up in nonlinear geometric optics, Ann. Sci. École Norm. Sup. 33 (2000) 132–209]. Then to get the usual sublinear growth on the hyperbolic part of the profiles, for every corrector, we consider , the space of bounded functions decomposing into a sum of pure transports and a “quasi compactly” supported part. We make a detailed analysis on the nonlinear interactions on which leads us to make a restriction on the set of resonant phases. We finally give a convergence result which justifies the use of “quasi compactly” supported profiles. [Copyright &y& Elsevier]

Abstract: This paper is concerned with Schrödinger equations whose principal operators are elliptic. Under certain degenerate condition we show the estimate of an oscillatory integral related to the solution operator in free case, and then employ fractionally integrated groups to obtain the estimate of solutions for the initial data belonging to a dense subspace of in the case of integrable potentials, which improves the corresponding result in [M. Balabane, H.A. Emami-Rad, estimates for Schrödinger evolution equations, Trans. Amer. Math. Soc. 292 (1985) 357–373]. [Copyright &y& Elsevier]

Abstract: In this paper, we study the Stokes system in the half-space , with . We give existence and uniqueness results in weighted Sobolev spaces. After the central case of the generalized solutions, we are interested in strong solutions and symmetrically in very weak solutions by means of a duality argument. [Copyright &y& Elsevier]

Abstract: This paper deals with a class of linear equations with boundary degeneracy. According to the degenerate ratio, the equations are divided into weakly degenerate ones and strongly degenerate ones, which should be supplemented by different Dirichlet boundary value conditions. After establishing some necessary existence, nonexistence and comparison principles, we investigate the optimal Hölder continuity of weak solutions in these two cases utilizing the Harnack inequality and the Morrey theorem, respectively. [Copyright &y& Elsevier]

Abstract: In this paper we study the existence, uniqueness and propagation of regularity to infinite order partial differential evolution equations. Our approach is essentially functional and brings interesting results even when we restrict ourselves to finite order equations. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a class of stochastic wave equations with nonlinear multiplicative noise. We first show that these stochastic wave equations generate random dynamical systems (or stochastic flows) by transforming the stochastic wave equations to random wave equations through a stationary random homeomorphism. Then, we establish the existence of random invariant manifolds for the random wave equations. Due to the temperedness of the nonlinearity, we obtain only local invariant manifolds no matter how large the spectral gap is unlike the deterministic cases. Based on these random dynamical systems, we prove the existence of random invariant manifolds in a tempered neighborhood of an equilibrium. Finally, we show that the images of these invariant manifolds under the inverse stationary transformation give invariant manifolds for the stochastic wave equations. [Copyright &y& Elsevier]

Abstract: In this paper, the authors study the equation in with . We first prove that for , the solution exists at least for a short time; then for , the existence and nonexistence of global (in time) solutions are studied in various situations. [Copyright &y& Elsevier]

Abstract: Suppose that is a constant and that is a continuous function with . This paper investigates N-dimensional singular, quasilinear elliptic equations of the form and gives some sufficient conditions for the equations to have (i) a decaying positive entire solution or (ii) infinitely many positive entire solutions each of which is bounded with a positive lower-bound. [Copyright &y& Elsevier]

Abstract: We let Ω be a smooth bounded domain of and a sequence of functions such that in . We consider a sequence of functions such that in Ω for all . We address in this paper the question of the asymptotic behavior of the ''s when . The corresponding problem in dimension 2 was considered by Brézis and Merle, and Li and Shafrir (among others), where a blow-up phenomenon was described and where a quantization of this blow-up was proved. Surprisingly, as shown by Adimurthi, Struwe and the author in [Adimurthi, F. Robert and M. Struwe, Concentration phenomena for Liouville equations in dimension four, J. Eur. Math. Soc., in press, available on http://www-math.unice.fr/~frobert], a similar quantization phenomenon does not hold for this fourth-order problem. Assuming that the ''s are radially symmetrical, we push further the analysis of the mentioned work. We prove that there are exactly three types of blow-up and we describe each type in a very detailed way. [Copyright &y& Elsevier]

Abstract: In this paper we continue the analysis of the blow-up of low energy sign-changing solutions of semi-linear elliptic equations with critical Sobolev exponent, started in [M. Ben Ayed, K. El Mehdi, F. Pacella, Blow-up and nonexistence of sign-changing solutions to the Brezis–Nirenberg problem in dimension three, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press]. In addition we prove axial symmetry results for the same kind of solutions in a ball. [Copyright &y& Elsevier]

Abstract: In this paper, assume that h is nonnegative and , we prove that if is sufficiently small, then there are at least three positive solutions of Eq. (1) in , where D is a bounded domain in . [Copyright &y& Elsevier]

Abstract: This paper is concerned with the robust stability for linear time-varying differential–algebraic equations. We consider the systems under the effect of uncertain dynamic perturbations. A formula of the structured stability radius is obtained. The result is an extension of a previous result for time-varying ordinary differential equations proven by Birgit Jacob [B. Jacob, A formula for the stability radius of time-varying systems, J. Differential Equations 142 (1998) 167–187]. [Copyright &y& Elsevier]